I'm trying to find out the double integral of the function over a unit disk. I have tried putting it in polar coordinated but that becomes too lengthy. Not sure how to move ahead.
Prove that $\displaystyle \frac\pi3 \leq \iint\limits_D \frac{dxdy}{\sqrt{x^2+(y-2)^2}} \leq \pi$ where $D$ is the unit disc.
You don't have to calculate the integral exactly because what is required is only lower bound. The statement is equivalent to $$\frac{1}{vol(D)}\int_D \frac{1}{\sqrt{x^2+(y-2)^2}}dxdy \geq \frac{1}{3}. $$So try first to show that $$ \frac{1}{\sqrt{x^2+(y-2)^2}}\geq \frac{1}{3}. $$(This is quite straightforward.)